Restrictions on Arrangements of Ovals of Projective Algebraic Curves of Odd Degree
نویسنده
چکیده
This paper investigates the first part of Hilbert’s 16th problem which asks about topology of the real projective algebraic curves. Using the Rokhlin-Viro-Fiedler method of complex orientation, we obtain new restrictions on the arrangements of ovals of projective algebraic curves of odd degree d = 4k + 1, k ≥ 2, with nests of depth k.
منابع مشابه
Real Plane Algebraic Curves with Asymptotically Maximal Number of Even Ovals
It is known for a long time that a nonsingular real algebraic curve of degree 2k in the projective plane cannot have more than 7k 2 2 − 9k 4 + 3 2 even ovals. We show here that this upper bound is asymptotically sharp, that is to say we construct a family of curves of degree 2k such that p k →k→∞ 7 4 , where p is the number of even ovals of the curves. We also show that the same kind of result ...
متن کاملArf Invariants of Real Algebraic Curves
. Let CA be the complex curve in CP(2) given by the same polynomial as RA. Thus RA = CA∩RP(2). RA is a M-curve precisely when CA \ RA consists of two punctured spheres which are interchanged by complex conjugation. Arbitrarily choose one of these components, say CA. The complex structure on CA induces an orientation on CA, and thus on each immersed circle of RA. Of course if we choose the other...
متن کاملPseudo-holomorphic and Algebraic Classifications
Almost all known restrictions on the topology of nonsingular real algebraic curves in the projective plane are also valid for a wider class of objects: real pseudo-holomorphic curves. It is still unknown if there exists a nonsingular real pseudo-holomorphic curve not isotopic in the projective plane to a real algebraic curve of the same degree. In this article, we focus our study on symmetric r...
متن کاملArrangements of an M-quintic with Respect to a Conic That Maximally Intersects Its Odd Branch
Under certain assumptions, the arrangements mentioned in the title are classified up to isotopy. Their algebraic realizability is discussed. §0. Introduction 0.1. Statement of main results. The connected components of the set of real points of a plane projective real curve are called branches. A branch is even (or an oval) if it is zero-homologous in RP. Otherwise it is odd (or a pseudoline). T...
متن کاملThe number of trees half of whose vertices are leaves and asymptotic enumeration of plane real algebraic curves
The number of topologically different plane real algebraic curves of a given degree d has the form exp(Cd + o(d)). We determine the best available upper bound for the constant C. This bound follows from Arnold inequalities on the number of empty ovals. To evaluate its rate we show its equivalence with the rate of growth of the number of trees half of whose vertices are leaves and evaluate the l...
متن کامل